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 Tanh

 http://functions.wolfram.com/01.21.20.0004.01

 Input Form

 D[Tanh[z], {z, n}] == Tanh[z] KroneckerDelta[n] + Sech[z]^2 KroneckerDelta[n - 1] - I^n n Sum[(((-1)^k Binomial[n - 1, k])/(k + 1)) Cosh[z]^(-2 k - 2) 2^(n - 2 k) Binomial[2 k, j] (k - j)^(n - 1) Sinh[(I n Pi)/2 + 2 (j - k) z], {k, 0, n - 1}, {j, 0, k - 1}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["Tanh", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Tanh", "[", "z", "]"]], RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], "n", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]]]], RowBox[List["k", "+", "1"]]], SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], "k"]], "-", "2"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["Sinh", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]"]], "2"], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "k"]], ")"]], " ", "z"]]]], "]"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n tanh ( z ) z n tanh ( z ) δ KroneckerDelta n + sech 2 ( z ) δ KroneckerDelta n - 1 - n n k = 0 n - 1 j = 0 k - 1 ( - 1 ) k cosh - 2 k - 2 ( z ) 2 n - 2 k ( k - j ) n - 1 k + 1 ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sinh ( n π 2 + 2 ( j - k ) z ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition z n z z KroneckerDelta n z 2 KroneckerDelta n -1 -1 n n j 0 k -1 k 0 n -1 -1 k z -2 k -2 2 n -1 2 k k -1 j n -1 k 1 -1 Binomial n -1 k Binomial 2 k j n 2 -1 2 j -1 k z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["Tanh", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Tanh", "[", "z", "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], " ", "n", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "2"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["Sinh", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]"]], "2"], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "k"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List["k", "+", "1"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29